On Mon, Feb 22, 2010 at 12:09 PM, Leigh J. Halliwell <[email protected]> wrote: > I'm quite profficient in the standard linear algebra with linear and > quadratic forms of random vectors. So the variance of an (m x 1) random > vector x is an (m x m) matrix whose ijth element is cov(x[i], x[j]), and > Var(Ax) = A Var(x) T(A). > > But I'd like to take the variance of an (m x n) random matrix X. This would > be an (m x n x m x n) array (or better, an ((m x n) x (m x n)) matrix) whose > (ij)(kl)th element is cov(x[ij], x[kl]). And I'd like also to generalize > that quardratic form. > > So I'm seeking to generalize matrices from {linear mappings from n-space to > m-space} to {linear mappings from (n1 X n2)-space to (m1 x m2)-space}. At > the very least, I can ravel (,) the axes and do it as (m1*m2) x (n1*n2) > matrices. But I wonder if something more powerful exists.
I have to admit that I am still lost. Someone like John Randall might be able to help you, but I am a complete novice when it comes to this kind of thing. I can perform experiments: X=: ? 5 5$0 Y=: ? 5 5 $0 E=:(+/%#)@, E ((-E)X) * ((-E)Y) _0.0339656 E ((-E)X) +/ .* ((-E)Y) 0.00833635 But I do not have the background to assign any meaningful concepts to these results. That said, I do have a question that might (or might not) help me: What do you have in mind for the expected values in the multi-dimensional case you have described? (Are you working with any dependencies?) -- Raul ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
